<h2>题目编号 : 74</h2>
<div style="color:#666;font-size:80%;">16 July 2004</div><br />
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<p>The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145:</p>
<p style='margin-left:50px;'>1! + 4! + 5! = 1 + 24 + 120 = 145</p>
<p>Perhaps less well known is 169, in that it produces the longest chain of numbers that link back to 169; it turns out that there are only three such loops that exist:</p>
<p style='margin-left:50px;'>169 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 363601 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 1454 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 169<br />
871 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 45361 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 871<br />
872 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 45362 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 872</p>
<p>It is not difficult to prove that EVERY starting number will eventually get stuck in a loop. For example,</p>
<p style='margin-left:50px;'>69 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 363600 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 1454 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 169 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 363601 (<img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 1454)<br />
78 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 45360 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 871 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 45361 (<img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 871)<br />
540 <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 145 (<img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> 145)</p>
<p>Starting with 69 produces a chain of five non-repeating terms, but the longest non-repeating chain with a starting number below one million is sixty terms.</p>
<p>How many chains, with a starting number below one million, contain exactly sixty non-repeating terms?</p>

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